Download Institute for Theoretical Physics of Phase Transitions

Phase Transitions
1
Phase Transitions
V. Dohm
Introduction
A phase, in the sense of equilibrium thermodynamics, is an equilibrium state of matter that is
spatially homogeneous on a macroscopic scale. Examples are the gas (vapor), liquid or solid
phases of matter, or the ferromagnetic and paramagnetic phases of a ferromagnet. Within the
solid state, a material may have several phases with different crystalline or magnetic structures.
For example, several different crystalline structures of ice exist at higher pressures. At low
temperatures, macroscopic quantum effects may manifest themselves in new phases of solids
and liquids such as the superconducting phases of certain metals and the superfluid phases of
Helium.
A phase transition of a macroscopic homogeneous system is a change of its equilibrium
state from one phase to another phase brought about by a change of thermodynamic variables,
e. g., the temperature T , the pressure P or the magnetic field H. Examples of phase transitions
that one encounters in daily life are boiling of water, condensation of vapor into water, and
melting of ice. Phase transitions are important in many industrial processes, for example the
solidification of molten metal. Further examples of fundamental physical significance are
magnetic phase transitions, superconducting and superfluid transitions, structural transitions
in solids and in liquid crystals. Analogies to phase transitions also exist in stationary states of
systems far from equilibrium such as the transition at the threshold of lasers and the RayleighBénard instability in hydrodynamics.
A phase diagram as a function of T and other “field” variables, e. g., P or H or the chemical
potential µ, shows the regions within which homogeneous equilibrium states exist. One example is the vapor-liquid-solid phase diagram of a pure (one-component) substance as a function
of T and P (Fig. 1). The three phases have different particle densities ρ. The one-phase regions are separated by coexistence curves at which the system may be in an inhomogeneous
equilibrium state. Two phases with different densities coexist in this state, e. g., water and ice
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Fig. 1: Phase diagram of a one-component substance.
Entry for list of contributors:
Volker Dohm
Department of Physics, Aachen
University, Aachen, Germany
P HASE T RANSITIONS
Revision date: 01-03-2005
2
Phase Transitions
Fig. 2: Phase diagram of a simple ferromagnet.
at 0 ◦C and at the pressure of one atmosphere. The three coexistence curves meet at the triple
point where three phases with three different densities coexist. The liquid-vapor coexistence
curve in the P–T phase diagram terminates at a critical point with a critical temperature Tc
at which the densities ρliquid and ρvapor become identical and beyond which no distinction
between liquid and vapor is possible.
An analogous example is the phase diagram of a simple ferromagnet as a function of the
external magnetic field H and temperature T (Fig. 2). In the paramagnetic phase above Tc
there is no magnetization at H = 0. The Curie temperature Tc corresponds to the critical
temperature in Fig. 1, and the line H = 0 below Tc corresponds to the liquid–vapor coexistence
curve. In the simplest case of a uniaxial ferromagnet where the magnetization M is aligned
parallel or antiparallel to a particular crystalline axis two ferromagnetic phases exist below Tc
corresponding to the liquid and vapor phases in Fig. 1. A spontaneous magnetization below
Tc with two different possible orientations in the limits H → 0+ and H → 0− corresponds to
coex
different densities ρcoex
liquid and ρvapor at the coexistence curve in Fig. 1.
Each substance has its specific phase diagram. Among the one-component substances,
Helium-4 and Helium-3 are exceptional since they do not solidify at low temperature but
become superfluid if the pressure is not too high. Fig. 3 shows the phase diagram of Helium-4
with a superfluid phase below a line Tλ (P) of critical points. The superfluid phase has unusual
properties such as frictionless mass flow. At a higher pressure a supersolid phase of Helium-4
exists where superfluid mass flow is observed in the solid state. Also dilute alkali gases may
undergo a transition to a superfluid state at very low temperatures under certain experimental
conditions.
For more-component systems (such as binary mixtures) a complete description of the equilibrium state requires the specification of the composition, e. g., of one concentration variable
in case of a binary mixture, in addition to the temperature T and the pressure P. The corresponding phase diagrams are more complex than that in Fig. 1.
In finite samples long-range forces may cause equilibrium states that are inhomogeneous
and depend on the shape of the sample. This is the case in magnetic systems with dipolar
forces as well as in metal-hydrogen alloys with an effective long-range interaction between the
protons. Inhomogeneous magnetic states with domains of different magnetic orientation may
also exist in a real magnetic material that is not a single-crystal with well-defined crystalline
axes.
A characteristic feature of a phase transition is its occurrence at sharply defined values of
Phase Transitions
3
(
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Fig. 3: Phase diagram of Helium-4.
the thermodynamic field variables. A counterexample is the transition of a gas to a plasma
state at very high temperatures which is not a phase transition but rather a gradual transformation. Also the glass transition is not an equilibrium phase transition with a uniquely defined
transition temperature but rather the solidification of an undercooled liquid in a metastable
state. Metastable states with very long life times may exist; a diamond, for example, is in a
metastable crystalline state of carbon whose equilibrium state is graphite which has a different
crystalline structure. In real experiments and in daily life phase transitions may not occur at
once but rather in the course of time through a nonequilibrium process, e. g., melting of ice or
boiling of water while the system is heated.
Thermodynamic properties and classification
The thermodynamic properties of a substance in thermal equilibrium are described by a free
energy F which, in the one-phase regions, is a smooth (analytic) function of its thermodynamic
variables. At the coexistence lines and at a critical point this function is non-analytic and the
observable thermodynamic quantities derived from F exhibit singularities.
Phase transitions are classified as being either first order or continuous (also called second
order) depending on the type of the singularities of the thermodynamic quantities at the transition. These singularities depend on the thermodynamic path in the phase diagram and on the
symmetry properties of the phases.
First-order phase transitions
A thermodynamic path across a coexistence line of Figs. 1 or 2 causes a phase transition that
is accompanied by a discontinuous change of the density ρ or of the magnetization M. Such
transitions are called first order since the discontinuities occur in the first derivatives (ρ and
M) of the free energy density. Examples of first-order transitions are melting, solidification,
sublimation and vaporization as well as many structural phase transitions in solids.
4
Phase Transitions
Fig. 4: P–ρ phase diagram of a one-component fluid.
For a one-component substance, a discontinuity of the particle density ρ = N/V corresponds to a discontinuity ∆V of the volume V = ∂F/∂P at fixed number N of particles. Similarly a discontinuity ∆S of the entropy S = −∂F/∂T exists at the coexistence curves of Fig. 1.
The jump ∆S is determined by ∆Q/T where ∆Q is the heat absorbed when one phase is transformed into the other phase. These discontinuities determine the slope of the coexistence
curves according to the Clausius–Clapeyron equation dP/dT = ∆S/∆V = ∆Q/T ∆V . In most
cases ∆Q and ∆V have the same sign, i. e., the slope is positive. An exception is the melting
curve of water which has a negative slope.
A line of coexistence in the P − T phase diagram corresponds to a region of coexistence
in the P − ρ phase diagram as shown for the example of the vapor phase and the liquid phase
in Fig. 4. Within the coexistence region no equilibrium state exists with a uniform density
coex
but the fluid is separated into macroscopic portions with different densities ρcoex
liquid and ρvapor .
The free energy of this inhomogeneous equilibrium state is lower than that of a state with a
homogeneous density.
In one-component systems the maximal number of coexisting phases is three (at the triple
point); in systems with r components it is r +2. If the number p of coexisting phases is smaller
than r + 2 then Gibbs’ phase rule states that it is possible to change r + 2 − p = f thermodynamic field variables (T , P or the chemical potentials) without losing one of the phases. f
represents the number of thermodynamic degrees of freedom of the multi-component system.
For one-component systems, f = 1 at the coexistence lines (p = 2) away from the triple point.
The corresponding phase diagram of a uniaxial ferromagnetic system as a function of the
magnetization M = −∂F/∂H is shown in Fig. 5. Here the two phases with positive and negative orientation of M have the same entropy which implies ∆S = 0 at the coexistence line
H = 0 with zero slope dH/dT = 0 in Fig. 3. A first-order transition with a discontinuous
change of the orientation of M occurs when the coexistence line is crossed by changing the
sign of H at fixed T below Tc . The jump of M implies a delta-function like singularity of the
susceptibility χ(T, H) = ∂M/∂H ∼ δ(H) at fixed T below Tc .
Magnetic and multicomponent solid systems also exist in which first-order transitions occur when the temperature is changed at fixed, zero ordering field H (in the case of structural
phase transitions, the “ordering field” is represented by externally applied forces causing dis-
Phase Transitions
5
Fig. 5: M–T phase diagram of a uniaxial ferromagnet.
tortions). Such transitions are also referred to as “temperature driven” phase transitions as
opposed to “field driven” transitions where H is varied at fixed T .
Continuous phase transitions and critical behavior
A continuous phase transition occurs along a thermodynamic path through the critical point
asymptotically parallel to the coexistence lines in Figs. 1 or 2. In this transition the density or
the magnetization vary continuously as the system is passing through the critical point. For
historical reasons, such transitions are also called second-order phase transitions. While the
first derivatives of the free energy are continuous the second derivatives, i. e., susceptibility,
compressibility, and specific heat, exhibit singularities (divergences in most cases) at Tc . A
continuous phase transition also occurs when a line of critical points is crossed, e. g., the λ-line
Tλ (P) of the superfluid transition in Helium-4, Fig. 3. A deep understanding of continuous
phase transitions and critical phenomena has been achieved due to the renormalization-group
theory.
A useful concept for the description of continuous phase transitions near a critical point is
Landau’s order parameter. It vanishes above the transition and increases smoothly from zero
at Tc to a finite value for T < Tc . For the uniaxial ferromagnetic transition the order parameter
is the spontaneous magnetization M which represents a measure for the magnetic order below
the critical temperature at H = 0. An analogous quantity below the gas-liquid critical point is
coex
coex
coex
the difference ∆ρ = ρcoex
liquid − ρvapor where ρliquid and ρvapor are the densities at the boundaries
of the coexistence region shown in Fig. 4. In terms of the relative temperature distance t =
(T − Tc )/Tc from Tc both order parameters vanish for t → 0− as M ∼ |t|β and ∆ρ ∼ |t|β where
β is called critical exponent. Other critical exponents γ and α appear in the critical behavior
of several thermodynamic quantities, such as the susceptibility χ of the ferromagnet and the
compressibility κ of the fluid or the specific heat C of both systems, which diverge as χ ∼
1/|t|γ and κ ∼ 1/|t|γ or C ∼ 1/|t|α as Tc is approached. An order parameter can be classified
according to the number of its components. For uniaxial ferromagnets and for the liquidgas transition M and ∆ρ are one-component order parameters. In isotropic magnetic systems
where crystalline-field effects are negligible near Tc the magnetization is a three-component
vector. In superfluids and superconductors the order parameter is a macroscopic quantummechanical wave function, i. e., a complex quantity with two components. There are also
magnetic systems where the magnetization lies in a plane perpendicular to a crystalline axis
6
Phase Transitions
corresponding to a two-component order parameter. In structural phase transitions of solids
(which may be continuous or discontinuous) the order parameter can have many components
depending on the number n of normal mode coordinates that are necessary to describe the
possible atomic displacements of the distorted phase.
Thermal fluctuations of the order parameter play an important role near Tc . For example
in a fluid one observes the phenomenon of critical opalescence near Tc , i. e., strong scattering
of light which is caused by increasing liquid-like and vapor-like regions created by correlated
density fluctuations. The range over which such fluctuations are correlated is called correlation length ξ. It increases as Tc is approached and can become comparable to or larger than
the wavelength of visible light. The strength of such correlations of order-parameter fluctuations at a spatial separation r is described by the correlation function G(rr). Above Tc it
decays exponentially ∼ e−r/ξ . Below Tc , the spatial decay of G(rr) is either exponential or
algebraic (power-law like) depending on the symmetry properties of the system (see below).
In a ferromagnet above Tc , ξ can be interpreted qualitatively as an average size of temporary
clusters of elementary magnetic moments with the same orientation. Macroscopic magnetic
order starts to exist permanently when ξ → ∞. This happens as T → Tc according to the power
law ξ ∼ 1/|t|ν with the critical exponent ν. The divergence of ξ is the physical origin of the
divergence of other thermodynamic quantities at a continuous phase transition. By contrast,
ξ is finite at a first-order phase transition of fluids and uniaxial magnets (but not of systems
with a continuous symmetry, see below). In anisotropic systems, e. g., magnets with layered
structures, two or three principle correlation lengths ξi = ξ0i /|t|ν may exist with different
amplitudes ξ0i for different directions i but with the same critical exponent ν for all directions. Right at Tc where ξ is infinite the correlations decay slowly according to the power
law G(rr) ∼ 1/|r|d−2+η where η is another critical exponent and d the dimensionality of the
system, e. g., d = 3 for ordinary bulk systems and d = 2 for films and layers. This decay is
isotropic for fluids but anisotropic for solids.
A unifying feature in the critical behavior of a continuous phase transition is the notion
of universality. While the value of Tc is nonuniversal, i. e., material dependent and different for each substance, the values of the critical exponents are universal, i. e., independent
of microscopic details of the interactions provided that the interactions are sufficiently shortranged and not strongly anisotropic. They have the same values for a large number of systems
within certain (n, d) universality classes which are classified according to the number n of
components of the order parameter and the dimensionality d of the system. For example, for
the (n = 1, d = 3) universality class the values of α ' 0.11 and β ' 0.33 are the same for
all fluids with different strengths of the van der Waals interaction and for all uniaxial ferromagnets and antiferromagnets in crystals with different strengths of the exchange interactions
and with different lattice structures. Thus a universality class includes both isotropic and
(weakly) anisotropic systems. Strongly anisotropic systems with different critical exponents
for different directions also exist (e. g., at Lifshitz points); these systems belong to a different
universality class. Furthermore, interactions of sufficiently long range change the type of critical behavior, such as dipolar interactions in magnetic systems. Such systems also belong to
a separate universality class. Although the long-range van der Waals forces in ordinary fluids
do not affect the critical exponents and the spatial density correlations in the regime r . ξ they
do affect their large-distance behavior (r & ξ) which becomes nonuniversal even close to Tc .
Apart from anisotropies with regard to the directions in coordinate space, crystalline-field
Phase Transitions
7
effects may yield an anisotropy in the space of the components of the order parameter (also
referred to as “spin anisotropy”). This may change the type of critical behavior. For example,
for an n-component order parameter with a cubic spin anisotropy the usual isotropic critical
behavior occurs as long as n < 4 but different critical exponents arise for larger values of n.
Another important feature of phase transitions is the concept of scaling. For example, the
equation of state M = M(T, H) of a ferromagnet which is usually a function of two variables
T and H, is reduced, near criticality, to
M/|t|β = A1 f (A2 H/|t|∆ )
(1)
where the scaling function f (x) depends only on the single scaled variable H/|t|∆ . Both the
function f (x) and the critical exponent ∆ are universal, only the scale factors A1 and A2 are
nonuniversal. Scaling forms like (1) exist also for the specific heat C(T, H); they imply exact
relations, so-called scaling laws, between the various critical exponents such that only two of
the critical exponents are independent. One important relation is 2 − α = d ν that can be tested
for the (n = 2, d = 3) universality class by measuring both the specific heat ∼ 1/|t|α and the
superfluid density ρs ∼ |t|ν near Tλ of Helium-4.
The feature of universality applies not only to the critical exponents but also to the amplitudes of the power laws of the thermodynamic quantities. There are universal ratios of these
amplitudes so that only two of these nonuniversal amplitudes are independent. For isotropic
systems and anisotropic systems with cubic symmetry this also includes the amplitude of the
correlation length ξ. Thus, once the universal quantities (critical exponents, scaling functions
and amplitude ratios) of a universality class are known the asymptotic critical behavior of very
different systems (e. g., fluids and magnets) is completely known provided that only the two
scale factors A1 and A2 are given. This property is called two-scale factor universality. However, it is not valid for anisotropic systems of non-cubic symmetry within a universality class
where additional information is necessary with regard to the amplitudes and orientation of the
correlation lengths ξi along the principle axes of the anisotropic system. Furthermore it is not
valid for the large-distance behavior of the correlation function G(rr) of ordinary fluids in the
regime r & ξ where the van der Waals forces imply a nonscaling form of G(rr).
The validity of universality and scaling of critical behavior has been proven by the renormalization-group theory (see below) which also predicts accurate values for the critical
exponents and other universal quantities. The superfluid transition of Helium 4 which belongs
to the (n = 2, d = 3) universality class is ideally suited for testing universality and the predictions of the renormalization-group theory. Here the experimental precision is exceptionally
high and experiments can be performed at various pressures along the “λ - line” Tλ (P) of critical points (Fig. 3). Universality implies that the critical exponents and amplitude ratios must
be independent of the pressure P. Existing experiments along the λ - line confirm universality
within the experimental resolution.
The most accurate value of a critical exponent is α = −0.0127 for the specific heat of
Helium-4 near Tλ (Fig. 6) obtained from an experiment which was conducted under microgravity conditions in space where inhomogeneous perturbations due to gravity are negligible.
This value agrees with the most accurate renormalization-group predictions.
8
Phase Transitions
Fig. 6: Specific heat of Helium-4 close to Tλ . The line shows the best-fit
function with the critical exponent α = −0.0127 [after J. A. Lipa et al., Phys.
Rev. B 68, 174518 (2003)].
Symmetry breaking at phase transitions
Phase transitions may or may not be accompanied by a change of symmetry properties. A
first-order transition from the liquid state to the crystalline state breaks the isotropy of the
fluid state. On the other hand, both the liquid and vapor phases of a fluid are homogeneous
and isotropic since these phases differ only with respect to the magnitude of particle densities.
Thus no particular change of symmetry properties appears in the first-order transition between
the vapor and liquid phases.
In a (temperature-driven) ferromagnetic transition at H = 0 the paramagnetic state with
no permanent magnetization above Tc turns into an anisotropic ferromagnetic state with a
particular orientation of the spontaneous magnetization M below Tc . This orientation is not
imposed by the internal ferromagnetic interaction or an external magnetic field, therefore this
phenomenon is called spontaneously broken symmetry. Within a statistical theory (see below)
H) for a system of finite volume V at finite H and
the symmetry is broken by first defining M (V,H
H). In real experiments the orientation of M is
then taking the limit M = limH→00 limV →∞ M (V,H
influenced by the initial and boundary conditions under which the ferromagnetic equilibrium
state is reached.
Two cases must be distinguished: continuous or discrete symmetry corresponding to an ncomponent order parameter with n ≥ 2 or n = 1, respectively. For example, the interaction of
an isotropic ferromagnet (n = 3) is invariant under arbitrary rotations of the magnetic variables
corresponding to a continuous symmetry. The interaction of a uniaxial ferromagnet (n = 1),
on the other hand, is invariant under a reversal of sign of the magnetic variables corresponding
only to a discrete symmetry. These symmetry properties imply characteristic differences at the
H = 0 coexistence line below Tc .
For a magnetic system with a rotationally symmetric interaction, all orientations of M are
energetically equivalent, thus there are no restoring forces against perturbations that cause a
finite rotation of the spontaneous magnetization in the absence of an external magnetic field.
This implies that the transverse and longitudinal susceptibilities have divergent contributions
∼ 1/|H| and ∼ 1/|H|(4−d)/2 , respectively, as the H = 0 coexistence line is approached for
H → 0 at fixed T < Tc . For uniaxial magnets such divergences do not exist, apart from the
Phase Transitions
9
contribution ∼ δ(H) to the susceptibility arising from the discontinuity of the order parameter.
There are also very weak, so-called essential singularities at the coexistence line.
At finite H below Tc the spatial dependence of the longitudinal and transverse correlation
functions is exponential ∼ e−r/ξ with finite correlation lengths ξ(T, H). For H → 0, ξ(T, H)
diverges as ∼ 1/|H|1/2 in isotropic magnets below Tc whereas it remains finite for uniaxial
ferromagnets. Correspondingly the spatial dependence of correlation functions at H = 0 below
Tc for systems with a continuous symmetry is a power law behavior similar to critical systems.
By contrast, it remains exponential at H = 0 for a uniaxial ferromagnet. However, the powerlaw behavior is not a true critical phenomenon but rather the effect of long-wavelength spinwave excitations (so-called Goldstone modes). In fluids and fluid mixtures such phenomena
do not exist and correspondingly the correlation lengths remain finite at the gas-liquid and at
the fluid-solid first-order transitions.
The Mermin–Wagner theorem states that there can be no spontaneous symmetry breaking
for systems with continuous symmetry in d ≤ 2 dimensions. Thus a finite critical temperature
Tc > 0 with a nonzero order parameter exists for d = 2 dimensional systems only if the order
parameter is one-component, e. g., in uniaxial magnetic systems but not for isotropic magnets
(n = 3). A special case are systems with a two-component order parameter such as superfluid
Helium-4. If such systems are two-dimensional (films) then a continuous phase transition exists, the so-called Kosterlitz–Thouless transition at a finite transition temperature TKT . Below
this there are long-range correlations of order-parameter fluctuations although there is no finite order parameter. The correlations result in a power-law behavior of G(rr) ∼ 1/|r|η with a
nonuniversal exponent η that depends on the temperature.
Another continuous symmetry that can be broken is the translational symmetry. This happens in the presence of an interface due to two-phase coexistence, e. g., of liquid and vapor.
Since, in the absence of external forces and of boundaries, there are no restoring forces against
a translation of an interface, there are soft long-wavelength interface fluctuations similar to
Goldstone modes in an isotropic ferromagnet. A localized interface cannot exist in systems
whose order parameter has more than one component, e. g., in an isotropic magnet, since the
order-parameter profile can change gradually between the different phases of that system by a
smooth orientational variation of the order parameter over macroscopic distances.
Statistical theory of phase transitions
A macroscopic system in equilibrium at finite temperature T exhibits thermal fluctuations of
local thermodynamic variables. At a phase transition these fluctuations are strongly correlated
and cause cooperative effects over macroscopic distances. The resulting singular behavior
of thermodynamic quantities can be calculated within the framework of statistical mechanics. The statistical properties of the fluctuations are described by the canonical distribution
exp (−H /kB T ) where kB is Boltzmann’s constant and H is the Hamiltonian of the system.
The basic link between microscopic degrees of freedom and macroscopic thermodynamic
quantities is provided by the partition function
Z = Tr exp(−H /kB T )
(2)
which is a sum over all microscopic states. The free energy is then obtained as F = −kB T ln Z.
10
Phase Transitions
Quantum versus classical theories
In many cases a classical Hamiltonian H is appropriate for the description of phase transitions,
such as the ordinary liquid-vapor or liquid-solid transitions. On the other hand, superconductivity and superfluidity are macroscopic quantum phenomena, and the existence of an order
parameter in the low-temperature phases of superconductors, superfluids, and magnets would
not be possible without quantum mechanics. The thermodynamic behavior in the vicinity of
a phase transition at finite temperatures, however, is determined not primarily by the value
of the order parameter but rather by order-parameter fluctuations. On a microscopic length
scale these fluctuations may be of quantum character but near a critical point with Tc > 0,
the asymptotic behavior is governed by order-parameter fluctuations of large clusters of size
ξ which attain a classical character for T → Tc , ξ → ∞, even for superfluids and superconductors. Therefore it suffices to use a classical model Hamiltonian H for the description of
fluctuation effects in the vicinity of continuous phase transitions at T > 0 (but not for quantum phase transitions at T = 0, see below). There is no such argument, however, for first-order
transitions with a finite correlation length.
Models
The feature of universality permits one to use the simplest possible model within a universality
class if one is interested only in universal properties. In a classical description, the partition
function does not depend on the kinetic energy term of H .
A magnetic phase transition can be modelled by the magnetic degrees of freedom σi
(“spins”) on the lattice points i of a rigid lattice. In the simplest case, the Ising model, each
spin can assume two values σi = 1 (“up”) and σi = −1 (“down”) which represent two possible
orientations of the elementary magnetic moments of a uniaxial magnet. The energy associated
with a configuration {σi } of spins is assumed to be
H = −J
∑
σi σ j
(3)
<i j>
where the sum is restricted to nearest-neighbor pairs of spins. A ferromagnetic coupling J > 0
between neighboring spins favors a parallel orientation whereas temperature-induced fluctuations tend to create a random orientation. Below a critical temperature Tc the spins are
predominantly parallel, resulting in two possible ordered states with a positive or negative
macroscopic magnetization M. The spins can also be coupled to an external magnetic field
H. The calculation of the partition function then yields the phase diagram of Fig. 2 and the
magnetization of Fig. 5.
By a transformation of the spin variables the model (3) can be used to describe a lattice gas
(where an up or down spin corresponds to the presence or absence of an atom) as realized,
e. g., for chemisorption at surfaces, or a binary alloy (where an up or down spin corresponds
to an atom of kind A or B). By a negative choice of J < 0 an antiferromagnet can be modelled
by Eq. (3). Different types of couplings (e. g., between next-nearest neighbors, or long-range
interactions) can be included. To cover the complete (n, d) universality classes, the model (3)
can also be generalized to the n-vector model where σi is replaced by an n-component vector
Si.
Instead of a lattice description, universality permits one to use a continuum description
in terms of a fluctuating “field” ϕ(xx) which represents a coarse-grained spin variable aver-
Phase Transitions
11
aged over a mesoscopic length scale. The corresponding Landau–Ginzburg–Wilson continuum Hamiltonian reads for isotropic d-dimensional systems with short-range interactions and
a scalar field ϕ(xx)
Z
r
1
H = d d x 0 ϕ2 + (∇ϕ)2 + u0 ϕ4
(4)
2
2
V
where r0 is linear in T − Tc and u0 > 0 is a nonuniversal coupling constant. Higher orders
in ϕ(xx) do not affect the universal properties of ordinary critical behavior. Eq. (4) can be
generalized to an n-component vector field ϕ (xx) and can be extended to include anisotropies,
surface effects, and long-range interactions.
An exact calculation of the partition function of these models in the bulk limit V → ∞ is
possible only in a few cases (e. g., for the d = 2 Ising model and for the d-dimensional n-vector
model in the limit n → ∞). In most cases it is necessary to use approximate analytic or numerical methods. Regarding isotropic two-dimensional systems, progress has also been achieved
by the method of conformal invariance. A systematic approach for general dimensions d is
provided by the renormalization-group theory.
Renormalization-group approach
The divergence of the correlation length ξ near a critical point causes a strong coupling among
infinitely many degrees of freedom. This implies that simple approximation schemes (such as
mean-field approximation) in the calculation of the partition function (2) become inapplicable. This problem is overcome by the renormalization-group (RG) approach. No unique RG
theory exists but rather different formulations of the RG appoach. Two major types can be
distinguished:
(i) Successive elimination of degrees of freedom: The trace in Eq. (2) is performed only
partially by summing over those degrees of freedom that are associated with short-wave
length fluctuations. This is repeated to generate a sequence of effective Hamiltonians
at larger and larger length scales which tend to a fixed-point Hamiltonian. Universality
is understood as the independence of the fixed-point Hamiltonian on the nonuniversal
parameters of the original model Hamiltonian. Different fixed-point Hamiltonians correspond to different universality classes. This approach can be performed in the space
of coordinates xi of lattice models such as Eq. (3) (real-space renormalization, block
spin methods) or in the space of wave vectors k of continuum models such as Eq. (4)
(Wilson’s momentum-shell renormalization). In both cases the result is a sequence of
decreasing correlation lengths relative to the remaining minimal length scale (or to the
maximal wave-number), which corresponds to a set of transformations of the critical system into a noncritical regime. These transformations have the mathematical properties of
a semigroup.
(ii) Field-theoretic RG approach: No degrees of freedom are eliminated but the parameters
and variables of the model are transformed in a single step provided that the model is
renormalizable in the field-theoretic sense. This is the case for the model (4) in d ≤ 4
dimensions. Renormalizability means that in the formal limit of an infinite wave-number
cutoff (corresponding to a vanishing of the lattice spacing), all “ultraviolet” divergences
12
Phase Transitions
can be absorbed by a reparameterization of r0 and u0 and of the field ϕ. These “renormalizations” define a renormalized theory whose universal critical behavior is the same
as that of the physical unrenormalized model. The existence of a renormalized theory
facilitates not only perturbative calculations (at infinite cutoff) but makes possible to circumvent the “infrared” (critical) divergences arising at small wave numbers and at Tc .
This is achieved by mapping the renormalized theory from the critical to the noncritical region where perturbation theory with respect to the renormalized coupling u of the
non-Gaussian ϕ4 term is applicable and where resummations of the perturbation results
are possible. The mapping involves an effective renormalized coupling constant u(`) satisfying an “RG flow equation” where ` ∝ |t|ν . This corresponds to a flow of effective
Hamiltonians of the formulation (i). Universality is explained by the independence of the
fixed-point value u∗ = lim`→0 u(`) on the bare coupling u0 .
The common property of both formulations (i) and (ii) is a mapping of the original problem
in the difficult region near criticality to a less difficult problem in a region well away from
criticality. An advantage of (i) is that it is well adapted to the determination of nonuniversal
features (like phase diagrams) whereas (ii) is most efficient in analytic calculations of universal properties (like critical exponents). The most accurate value of the critical exponent α
(obtained from a renormalized perturbation theory within the model (4) up to seventh order
in u∗ ) agrees with the most accurate experimental value of α (at the superfluid transition of
Helium-4, see Fig. 6) within the theoretical and experimental error bars.
The RG approach can also be applied to first-order transitions, to dynamic properties at
phase transitions, to confined systems, to quantum phase transitions, and to phase transitions
in elementary particle physics. It can also be implemented in numerical methods. The RG
approach can even be used in problems outside the area of phase transitions where the coupling
of degrees of freedom on many length scales is important (e. g., in nonlinear dynamics).
Dynamics of phase transitions
The dynamics of phase transitions deals with three kinds of phenomena in a macroscopic system near a phase transition: (i) the time-dependent properties of a nonequilibrium state that
approaches the equilibrium state, (ii) the transport properties in a stationary state close to the
equilibrium state, and (iii) the time and frequency dependence of correlated fluctuations in the
thermal equilibrium state. Two examples of case (i) are (a) the relaxation of the order parameter from a nonequilibrium value towards its spontaneous equilibrium value near Tc and (b) the
phase separation in binary mixtures from a homogeneous state to the equilibrium state with
two coexisting phases below Tc . Examples of the cases (ii) and (iii) are the heat conduction
in a fluid near Tc in the presence of a small stationary heat current, and the frequency dependence of the dynamic structure factor of magnetic systems or of fluids in thermal equilibrium
as measured by neutron or light scattering experiments. Such dynamic phenomena can be
studied also by computer simulations.
Even in thermal equilibrium, a classification of dynamic phenomena near phase transitions
is significantly more complex than that of static phenomena. While the latter are determined
entirely by the canonical equilibrium distribution exp(−H /kB T ) the dynamic phenomena depend, in addition, on the hydrodynamic variables coupled to the order parameter. They also
depend on whether the order parameter itself is a hydrodynamic variable (like the mass density
Phase Transitions
13
of a fluid whose total mass is conserved), or a non-conserved quantity (like the sublattice magnetization of an antiferromagnet or the macroscopic wave function of superfluid Helium 4).
The corresponding equations of motion depend on dissipative parameters (relaxation rates,
diffusion constants) and on non-dissipative couplings (e. g., strength of convective terms in
fluids) which do not appear in the Hamiltonian H of the equilibrium distribution. Typical dissipative effects are relaxation, damping and diffusive processes. Examples for non-dissipative
dynamic phenomena are propagating modes (sound waves, spin waves) and convective transport of heat and mass.
A large variety of dynamic phenomena exists near phase transitions. Only a few examples
shall be considered in the following.
Critical dynamics
An elementary case of a dynamic critical phenomenon is the relaxation of the order parameter
towards its equilibrium value near Tc if the order parameter is a non-conserved quantity (such
as in uniaxial antiferromagnets or in superfluid Helium-4). At T 6= Tc there is a decay from
a non-equilibrium initial value towards the equilibrium value according to an exponential law
∼ exp(−t/τ) for long times t, with a relaxation time τ. The latter increases as T approaches
Tc according to the power law τ ∼ ξz where z is the universal dynamic critical exponent and
ξ is the correlation length. This phenomenon is called critical slowing down. It is understood
qualitatively as a consequence of a slowing down of the motion of large correlated clusters of
size ξ as T → Tc . This implies that the long-wavelength fluctuations of the order parameter
become slow variables near Tc even if the order parameter is not a hydrodynamic variable.
Right at Tc the relaxation is no longer exponential but a power law ∼ 1/t y for large times t,
with the critical exponent y = β/νz. Universal dynamic critical behavior may also occur on
short time scales.
In isotropic ferromagnets the total magnetization is conserved which implies that above
Tc the local magnetization undergoes a diffusion process. Close to Tc the diffusion constant
behaves as D ∼ ξ2−z with a universal dynamic critical exponent z = (5 − η)/2. Here the
critical slowing down manifests itself by the vanishing of the diffusion constant D → 0 for
T → Tc .
Another example for a striking dynamic critical effect is the divergence of the thermal
conductivity at the critical point of ordinary fluids and at the superfluid transition of Helium-4.
On the basis of the renormalization-group theory, the concepts of universality and scaling
of static critical phenomena can be extended also to critical dynamics. The dynamic critical
exponent z is universal within a dynamic universality class which, however, consists of a
significantly smaller class of systems than in statics. For example, the static critical exponents
(α, β, . . . ) of uniaxial magnets and of ordinary fluids are the same but their dynamic critical
exponents z are quite different because the equations of motion for magnets and fluids have a
different structure.
Dynamics of first-order transitions
According to Fig. 4, homogeneous equilibrium states cannot exist inside the coexistence region. An initially homogeneous nonequilibrium state in this region will decay into a phaseseparated state. Near the boundary of this region, however, the system can remain in a homogeneous state over a finite period of time. This phenomenon is called metastability. In a fluid
14
Phase Transitions
Fig. 7: T –ρ phase diagram of a fluid. The arrows indicate quenches into
metastable and unstable states within the coexistence region.
metastability manifests itself in the form of an overheated liquid and of an undercooled (supersaturated) vapor. The corresponding regions of metastabilility are indicated schematically
in the temperature-density phase diagram of Fig. 7. The life times of such metastable states
may range from very short to long times, depending on the type of system. Due to the effect
of critical slowing down, life times can be increased if the temperature T . Tc is close to a
critical point.
The process of phase separation, i. e., the evolution in time from a homogeneous initial
state towards the inhomogeneous equilibrium state with two coexisting phases, can be initiated
by abruptly cooling a fluid from a homogeneous vapor state at fixed density ρinitial (i. e., in a
sealed container) down into the two-phase region of the phase diagram of Fig. 7. Two cases
must be distinguished.
(i) If the quenched state is close to the boundary of the single-phase regions, e. g., in the
metastable region near ρcoex
vapor , a spontaneous formation of droplets of various sizes occurs
via nucleation. The droplet size may grow or shrink depending on the radius of the
droplet. Beyond a critical radius the droplet size grows and domains of liquid with the
equilibrium density ρcoex
liquid will occupy a fraction of the container.
(ii) If the quenched state is put further within the coexistence region, e. g., at the critical
density ρc , the state is unstable with regard to spontaneous long-wavelength fluctuations.
Thus spatial density fluctuations even with very small amplitudes grow towards a sepacoex
ration of two macroscopic phases with different equilibrium densities ρcoex
vapor and ρliquid .
The latter process is called spinodal decomposition.
In Fig. 7 the dashed line indicates the so-called “spinodal curve” which, within approximate
classical theories, plays the role of a boundary between metastable and unstable regions. This
boundary, however, is not sharp (except for systems with long-range interactions), there is
only a gradual transition between the processes of nucleation and spinodal decomposition in
the region around the spinodal curve. Similar processes occur in the phase separation of a
Phase Transitions
15
Fig. 8: Specific heat data of Helium-4 near Tλ in confined geometries (film,
channel, box) with L = 1 µm. The line shows the bulk specific heat, compare
also Fig. 6 [after M. O. Kimball et al., Phys. Rev. Lett. 92, 115301 (2004)].
fluid mixture of two kinds of atoms A and B with two different equilibrium densities ρA and
ρB of the A-rich and B-rich phases. Spinodal decomposition can also be observed in various
other systems such as polymers, alloys, and colloidal systems.
Confined systems
Sharp singularities of the thermodynamic quantities at phase transitions can occur only in infinite systems. These singularities are modified if some or all dimensions of the system are
finite. In real finite systems these singularities are rounded. For macroscopic systems such
finite-size effects only exist very close to the transition and are often beyond experimental resolution, partly because they are masked by rounding effects due to gravity, impurities etc. If,
however, the characteristic size of the confining lengths is mesoscopic or microscopic, such
effects are measurable, for example in films, channels, and boxes filled with Helium-4 near
the superfluid transition as shown in Fig. 8. Furthermore, computer simulations of phase transitions are necessarily restricted to small model systems where finite-size and surface effects
become appreciable. The typical range in which such effects become significant is characterized by the correlation length ξ. The order-parameter fluctuations “feel” the confining length
L when the system is so close to the transition such that ξ & O(L).
In general, physical quantities of finite systems depend not only on the system size L and
16
Phase Transitions
the correlation length ξ but also on other lengths, e. g., on the range `0 of the van der Waals
interaction if the system is a fluid, and on the lattice spacing a0 if the system is a crystal. The
finite-size scaling hypothesis predicts a considerable simplification near a continuous phase
transition: for large L, the L dependence enters only in the form of the ratio L/ξ but not in
the form of L/a0 or L/`0 . For example the ferromagnetic susceptibility χ(T, L) which is a
function of the two variables T and L above Tc is predicted to have the scaling form
χ/ξγ/ν = Aχ φ(L/ξ)
(5)
where the scaling function φ(x) depends only on the single scaled variable x = L/ξ and where
φ(x) has certain universal properties. The same structure is predicted to be valid for the compressibility κ(T, L) of a confined fluid. Eq. (5) implies that at Tc the susceptibility of a finite
ferromagnet has a power-law dependence on the length L as ∼ Lγ/ν for large L. On the basis of
this property it is possible to use numerical data of finite model systems to obtain information
about the universal critical exponents of infinite systems.
Finite-size scaling functions such as φ(x) are less universal than bulk scaling functions,
such as f (x) of the bulk magnetization (1), because they depend on the geometry of the confined system as well as on the boundary conditions for the order parameter. Furthermore they
depend on whether the system is intrinsically isotropic (like fluids) or intrinsically anisotropic
(like solids). For the latter systems, finite-size and surface effects depend on the orientation of
the confining surfaces with respect to the orientation of the anisotropy axes. Furthermore, even
in isotropic fluids the finite-size effects become dependent on the van der Waals interaction in
the range L & O(ξ).
Finite-size scaling can also be applied to first-order transitions. Here the susceptibility
of Ising-like systems behaves as ∼ Ld for large L at fixed T < Tc in contrast to ∼ Lγ/ν at
Tc . Finite-size effects near first-order transitions of systems with a finite correlation length ξ
(fluids, uniaxial magnets, etc.) can be described within the standard Gaussian theory of thermodynamic fluctuations. For systems with a broken continuous symmetry (isotropic magnets,
superfluids, etc.) which have an effectively infinite correlation length the theory is complicated
by the existence of Goldstone modes (spin waves).
Confined systems have inhomogeneous density profiles near the boundaries. Within a
boundary layer critical phenomena can occur that belong to universality classes different from
the d = 3 bulk universality class. For example, if the exchange couplings at the surface of a
magnetic system are sufficiently strong a surface magnetization may exist below a surface critical temperature Tc,surface that is well above the bulk transition temperature Tc,bulk . If then the
temperature is further decreased the surface undergoes another transition, a so-called extraordinary transition, at Tc,bulk . Other phase transitions at boundaries are the wetting transition at
a triple point and the roughening transition at the surface of Helium-4 crystals. Finite-size and
boundary effects can also be observed for dynamic quantities such as the thermal conductivity
of confined Helium-4 near the superfluid transition.
Quantum phase transitions
At T > 0 the stability of a phase is connected to a minimum of the free energy which results
from a competition between internal energy and entropy. At T = 0, all thermal fluctuations
are frozen out and it is only the minimum of the energy that determines which phase of a
Phase Transitions
17
Fig. 9: Phase diagram of the quantum Ising model in a transverse magnetic
field h. The dot at hc is a quantum critical point.
macroscopic system is stable. A quantum phase transition of a macroscopic system at T = 0
is a change from one ground state to another ground state, due to a variation of the relative
strength of an internal interaction parameter g which is caused by a change of externally
controllable variables such as the pressure or the magnetic field. The ground state energy is
a singular function of g at g = gc where the transition occurs. In particular near a continuous
quantum phase transition there are long-range spatial correlations on the scale of a correlation
length ξ which diverges as ξ ∼ 1/|g − gc |ν with a critical exponent ν. The behavior near gc is
entirely governed by quantum fluctuations rather than thermal fluctuations.
One example of a quantum phase transition is the transition of a uniaxial ferromagnet
from its ferromagnetic ground state at T = 0 to a paramagnetic ground state by applying an
external magnetic field transverse to the magnetic axis. This is illustrated in the phase diagram
(Fig. 9) of a quantum Ising model with an exchange coupling J in the presence of a transverse
magnetic field h = Jg. The model Hamiltonian reads
H = −J
∑
<i j>
σzi σzj − h ∑ σxi
(6)
i
where σzi and σxi are the non-commuting components of the spin- 21 operator σ i at the lattice
site i. For h = 0, Eq. (6) corresponds to the ordinary Ising model (3) which has a classical
phase transition with a finite critical temperature and a ferromagnetic phase below Tc . If the
field h is increased at finite T < Tc the ferromagnetic order is destroyed at the line of classical
critical points Tc (h) where the critical exponents are the same as at Tc = Tc (0). If, however,
h is increased at T = 0 a critical value hc is reached that corresponds to a quantum critical
point. The critical exponents at this quantum phase transition are different from those at the
line Tc (h) of critical points. Quantum critical behavior is also observed if the temperature T is
lowered at fixed critical value hc . Away from hc and at T > 0 there is a competition between
quantum and thermal fluctuations.
A fundamental difference between classical and quantum critical behavior is the fact that in
a quantum description all equilibrium properties depend on the dynamics. The density operator e−H /kB T can be considered formally as a time evolution operator e−iH τ/h̄ in the imaginary
time interval τ = −ih̄/kB T . In the partition function the time formally plays the role of an
18
Phase Transitions
extra dimension in addition to the d spatial dimensions of the quantum system. In some cases
this implies a direct analogy between the critical behavior of a d-dimensional quantum system
and that of a (d + 1)-dimensional classical system. This is indeed the case for the 1-, 2-, or
3- dimensional quantum Ising model with the Hamiltonian (6) whose critical exponents at the
quantum critical point are those of the 2-, 3-, or 4- dimensional classical Ising model, respectively. At finite T the extra time dimension is finite in extent and corresponds formally to a
classical system in d + 1 dimensions in a slab geometry with finite thickness h̄/kB T .
A further example of a quantum phase transition is the transition of ultracold bosonic particles (Alkali atoms) in an optical lattice from a superfluid phase (with a long-range phase
coherence of the macroscopic quantum-mechanical wave function) to a Mott insulator phase
(where the atoms are localized at the lattice sites without phase coherence across the lattice). This is brought about by changing the potential depth of the lattice. A large variety of
quantum phase transitions also exists in correlated electron systems such as superconductors,
heavy-fermion compounds, and Fermi liquids.
Other phase transitions
Multicritical points
A transition point between two phases may become a line of transition points if some parameter p of the system is varied continuously. Examples of lines of critical points are shown in
Figs. 3 and 9. A so-called multicritical point occurs if the type of critical behavior changes
abruptly at a special value pc of the parameter p. It is called a tricritical point if the transition
line beyond pc becomes a line of first-order transitions. This can be observed, for example, in
antiferromagnets or metamagnets (where a magnetic field plays the role of p) and in ternary
and quaternary fluid mixtures (where the concentration can be varied). It is also observed in
mixtures of liquid Helium-4 and Helium-3 where the Helium-3 concentration c3 plays the role
of p. Here the continuous superfluid transition turns into a first-order transition at higher c3
where the mixture is separated into superfluid and normal-fluid phases. A bicritical point separates two lines of continuous transitions which belong to two different universality classes.
For example, an anisotropic antiferromagnet can have a uniaxial (n = 1) order parameter for
small magnetic fields H which, for larger H, is flipped into a two-component order parameter
perpendicular to the anisotropy axis.
So-called incommensurate phases may exist, e. g., in magnetic systems with competing
(ferromagnetic and antiferromagnetic) interactions. The magnetic order is commensurate with
the underlying lattice if the magnetic structure has spatial periods whose ratios with the lattice
constant are integer or rational numbers; otherwise the magnetic order is called incommensurate. The period of the structure of the ordered phase can be characterized by a wave vector
q . There are systems with a transition from a commensurate to an incommensurate phase
when an external parameter p is varied at fixed temperature T . In the incommensurate phase,
q (T, p) may vary continuously as a function of p. The Lifshitz point of a magnet is a multicritical point where incommensurate, commensurate, and paramagnetic phases meet. Near
the Lifshitz point the system is strongly anisotropic in the sense that not only the amplitude of
the correlation length but also the critical exponents ν and η depend on the direction. There
are also structural phase transitions in alloys exhibiting commensurate-incommensurate transitions and Lifshitz points.
Phase Transitions
19
Percolation transition
Real systems may have impurities, such as non-magnetic atoms in a magnetic material. Random impurities can change or even suppress the critical behavior. If the impurity concentration
p is sufficiently high the magnetic order is destroyed. The critical temperature Tc (p) of a dilute magnet is lowered with increasing p and vanishes at a critical value pc . If p is increased
at T = 0 a phase transition occurs at pc . Above pc the magnetization vanishes. As p is lowered from above pc , a percolating infinite cluster of correlated magnetic atoms is formed for
the first time at pc . This percolation transition at pc has much in common with a critical
phenomenon, with p − pc being the analog of T − Tc .
Phase transitions in elementary particle physics
Ordinary phase transitions in the area of condensed matter physics occur at temperatures between 10−5 K, e. g., Bose–Einstein condensation in dilute gases, and 103 K, e. g., melting of
solids. In the area of elementary-particle physics, two other types of phase transitions at extremely high temperatures are considered: the electroweak transition at Telweak ' 1015 K and
the deconfinement transition of QCD (quantum chromodynamics) at TQCD ' 1012 K. These
transitions, or at least associated sharp crossovers, are believed to have occurred in the early
universe during the extremely fast cooling process after the big bang. In the electroweak transition, the unification of electromagnetic and weak interactions was removed below Telweak
and particles (e. g., vector bosons and quarks) acquired mass via the Higgs mechanism. In the
QCD phase transition, the quark-gluon plasma was cooled down to the hadronic phase below
TQCD where quarks were confined to hadrons. These phase transitions are related to symmetry breaking in certain limiting cases of theoretical models. Collective phenomena related to
the confinement–deconfinement transition are studied experimentally in heavy-ion collisions
at ultrarelativistic energies. Furthermore, different phases of hadronic matter at high density
(e. g., color-superconducting phases) may exist in the interior of neutron stars.
Nonequilibrium systems
Phase-transition like phenomena are not restricted to thermal equilibrium states but also exist
in stationary nonequilibrium states, e. g., due to shear flow or heat flow. Examples are the
critical behavior of ordinary liquids as well as the phase separation in binary fluids and polymeric solutions in the presence of shear flow, or the superfluid transition of Helium-4 in the
presence of a finite heat current. There is also a large variety of transitions that occur only far
from equilibrium due to the presence of a steady energy flow. One example is the transition of
laser light at a critical pump power from a state with uncorrelated photons of non-macroscopic
intensity to a state of coherent light of macroscopic intensity. In many such transitions a spatially uniform state is changed into a non-uniform state with spatial variation. Examples for
such a pattern formation are (i) the Rayleigh–Bénard instability in hydrodynamics where a
fluid layer is heated from below and where the uniform diffusive heat transport in the presence of a small temperature gradient turns into a convective motion with a roll pattern above
a critical Rayleigh number; (ii) electrohydrodynamic convection in nematic liquid crystals
where hydrodynamic flow (convection rolls) can be induced by applying an electrical field;
(iii) the Taylor–Couette flow in a fluid between two coaxial cylinders where the inner cylinder
is rotated and the outer cylinder is at rest or rotated in the reverse direction: the concentric
flow for small rotation speed turns into a flow pattern with vortices (circulating rolls) above a
20
Phase Transitions
critical rotation speed; other patterns with travelling rolls can occur at higher rotation speed;
(iv) the onset of spatially dependent chemical reactions or even of spatio-temporal oscillations
in certain mixtures of reactants; (v) instabilities and generation of current filaments in semiconductors under strong excitation conditions such as high electric fields and high-current
injection.
The concepts of phase transitions are also applied to collective phenomena outside physics,
e. g., in studies of traffic flow or in models of sociology.
See also: Critical Points; Crystal Growth; Dynamic Critical Phenomena; Equations of
State; Ferroelectricity; Ferromagnetism; Helium, Liquid; Ising model; Liquid
Crystals; Magnetic Ordering in Solids; Metal–Insulator Transitions; Order–Disorder
Phenomena; Renormalization; Superconductivity Theory; Thermodynamics,
Equilibrium; Thermodynamics, Nonequilibrium
&
.
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